The exercise statement (roughly): Assume there is a terrorist prevention system that has a 99% chance of correctly identifying a future terrorist and 99.9% chance of correctly identifying someone that is not a future terrorist. If there are 1000 future terrorists among the 300 million people population, and one individual is chosen randomly from the population, then processed by the system and deemed a terrorist. What is the chance that the individual is a future terrorist?

Attempted exercise solution:

I use the following event labels:

A -> The person is a future terrorist

B -> The person is identified as a terrorist

Then, some other data:

$$P(A)=\frac{{10}^{3}}{3\cdot {10}^{8}}=\frac{1}{3\cdot {10}^{5}}$$

$$P(\overline{A})=1-P(A)$$

$$P(B\mid A)=0.99$$

$$P(\overline{B}\mid A)=1-P(B\mid A)$$

$$P(\overline{B}\mid \overline{A})=0.999$$

$$P(B\mid \overline{A})=1-P(\overline{B}\mid \overline{A})$$

What I need to find is the chance that someone identified as a terrorist, is actually a terrorist. I express that through P(A | B) and use Bayes Theorem to find its value.

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{P(B\mid A)\cdot P(A)}{P(B\mid A)\cdot P(A)+P(B\mid \overline{A})\cdot P(\overline{A})}$$

The answer I get after plugging-in all the values is: $3.29\cdot {10}^{-3}$, the book's answer is $3.29\cdot {10}^{-4}$.

Can someone help me identify what I'm doing wrong? Also, in either case, I find that it is very unintuitive that the probability of success is so small. If someone could explain it to me in more intuitive terms I'd be very grateful.

Attempted exercise solution:

I use the following event labels:

A -> The person is a future terrorist

B -> The person is identified as a terrorist

Then, some other data:

$$P(A)=\frac{{10}^{3}}{3\cdot {10}^{8}}=\frac{1}{3\cdot {10}^{5}}$$

$$P(\overline{A})=1-P(A)$$

$$P(B\mid A)=0.99$$

$$P(\overline{B}\mid A)=1-P(B\mid A)$$

$$P(\overline{B}\mid \overline{A})=0.999$$

$$P(B\mid \overline{A})=1-P(\overline{B}\mid \overline{A})$$

What I need to find is the chance that someone identified as a terrorist, is actually a terrorist. I express that through P(A | B) and use Bayes Theorem to find its value.

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{P(B\mid A)\cdot P(A)}{P(B\mid A)\cdot P(A)+P(B\mid \overline{A})\cdot P(\overline{A})}$$

The answer I get after plugging-in all the values is: $3.29\cdot {10}^{-3}$, the book's answer is $3.29\cdot {10}^{-4}$.

Can someone help me identify what I'm doing wrong? Also, in either case, I find that it is very unintuitive that the probability of success is so small. If someone could explain it to me in more intuitive terms I'd be very grateful.